Let Ω \subset \mathbb{R}^{n} be a domain and let α < n − 1. We prove the Concentration-Compactness Principle for the embedding of the space W_{0}^{1}L^{n} log^{\alpha } L(Ω) into an Orlicz space corresponding to a Young function which behaves like (t^{n/n-1-\alpha }) for large t. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem 1.6 where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula P: = \left( {1 - \left\| {\Phi (|\nabla u|)} \right\|_{L^1 (\mathbb{R}^n )} } \right)^{ - 1/(n - 1)} ., Robert Černý., and Obsahuje seznam literatury
We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.